76
S. Goto and H. Hino
the function I
eq
f does not induce a metric tensor field with respect to the coordinates
p
f
eq .
Although I
eq
f does not induce a metric tensor field with respect to p
f
eq , the function
I
eq
f induces a metric tensor field with respect to p eq . Then its dual coordinate system
is induced. For the sake of completeness, this is briefly discussed below.
On the manifold R
|Γ | , let p eq be a coordinate system, I
f
eq a function of p eq , and
define
g
Γ
f :=
j∈Γ
k∈Γ
∂
2 I
eq
f
∂ p
j
eq ∂ p k
eq
d p
j
eq ⊗ d p
k
eq .
If f is chosen as f ( p
j
eq ) = ln p
j
eq , ( j ∈ Γ ) so that
I
eq
f ( p eq ) =
j∈Γ
p
j
eq ln p
j
eq ,
then one has the Shahshahani metric tensor field [33, 40]:
g
S
:=
j∈Γ
1
p
j
eq
d p
j
eq ⊗ d p
j
eq .
The dual coordinates for this choice of f , f ( p
j
eq ) = ln p
j
eq , are
P
j
:= p
j
eq , and P
∗
j :=
∂ I
eq
f ( p eq )
∂ p
j
eq
= 1 + ln p
j
eq ,
j = 1, . . . , |Γ |,
in the sense that
g
S
∂
∂ P j ,
∂
∂ P
∗
k
= δ
k
j .
Notice that if f ≡ 1, then I
eq
f is not convex. Thus in this case Riemannian metric
tensor field is not induced. In addition, in this case p
f
eq = p eq ∈ S
|Γ |−1 .
4.5.2 Geometry of Nonequilibrium States
So far geometry of equilibrium states has been discussed. One remaining issue is
how to give the physical meaning of the set outside A I
eq
f
, C
Γ
f \ A I
eq
f
. A natural
interpretation of C
Γ
f \ A I
eq
f
would be some set of nonequilibrium states. We make
this interpretation in this paper (see also [21]).
In the contact geometric framework of thermodynamics, the equilibrium state
is identified with a Legendre submanifold. Then, as found in [3] and [21], some